How should one picture a topology/ topological space?

I can form a mental image of sets with structures like metrics or norms. But if I try to picture a topology/ topological space I fail every time. The information provided in Wikipedia confuses me quite a bit since the concept of topology is new to me.

So, is there a (preferably simple) explanation of a topology placed on a set? And how should someone picture such topologies/ topological spaces?

Solution 1:

Topological space come in very different flavors and therefore I don't think that there is one good mental picture to provide the general idea behind the concept of topological spaces.

Apart from the usual examples of a topological spaces like $\mathbb R^n$ with its metric, there are more exotic ones.

For example:

• Let $X \subseteq \mathbb N$ be open iff $X \in \{\emptyset, \mathbb N \}$ or $X = \{1,2, \ldots, n \}$ for some $n \in \mathbb N$. This defines a topology.
• Let $2^\mathbb N = \{ (a_n)_{n \in \mathbb N} \mid a_n \in \{0,1\} \text{ for all } n \in \mathbb N \}$ be the set of infinite 0 1 sequences. Then $$ d: 2^\mathbb N \times 2^\mathbb N \rightarrow \mathbb R, \ \left( (a_n)_n , (b_n)_n \right) \mapsto \frac {1}{ \min \{n \mid a_n \neq b_n \}} $$ defines a metric and thus a topology on $2^\mathbb N$. This topology is compact and zero-dimensional which gives it a very different flavor than the usual topology on $\mathbb R$.
• For a fixed language $L$ we can define a compact topological space on the set of $\mathcal T$ of all $L$-Theories which gives rise to a nice proof of the famous compactness theorem (see here).
• Another example is the Zariski-Topology on the spectrum of a fixed commutative ring, which is studied in classical algebraic geometry.
• ...

It might be a good idea to play a little with some simple examples of topological spaces (including some "weird" ones) to get used to this definition.

Solution 2:

If you start with a metric space $X$, then you define the open sets in $X$ to be those $U\subseteq X$ satisfying that for all $x\in U$ there exists $\varepsilon >0 $ such that $B_\varepsilon (x)\subseteq U$. The reason we are interested in open sets (and their complements, known as closed sets) is due to various elementary theorems of metric spaces which I assume you've seen (otherwise I will just say that the open sets allow to precisely capture the notion of continuity). It is quite easy to prove that the open sets in a metric space are closed under finite intersections and under arbitrary unions.

Now, by analysing proofs one discovers that some of the theorems we like can actually be stated in terms of open/closed sets only, and the proof can be obtained only by using the properties of open sets mentioned above. That means that if one is in a situation that you start with a metric space, note what the open sets are, and then suffer from amnesia whereby you totally forget what the metric was, you can still do quite a lot with the remaining 'space'. Of course, amnesia is not the reason for this, rather we realise that the metric information can be forgotten as long as one remembers the open sets, at least for some purposes (loosely speaking, those related to continuity). So at this level it's a game - there is a metric, but we pretend to only have the open sets it produces, and we see what we can still do with it.

Now, we change the setting. What if there really is no metric at all, only a bunch of subsets which are closed under finite intersections and arbitrary unions? Well, then this is a topology and the resulting thing is a topological space, which you can think of as the result of forgetting some metric that led to these open sets. This turns out to be extremely useful.

Now here is the nice thing. Any topological space is in fact coming from a metric, if one slightly generalises what metric means. The details of this is in Flagg's paper "Quantales and continuity spaces". The bottom line then is that any topology arises as the collection of open sets for some $V$-valued metric space. So you can really think of a topology as the result of temporary amnesia, forgetting the metric.